Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains

This paper is concerned with an explicit value of the embedding constant from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W^{1,q}(\Omega)$\end{document}W1,q(Ω) to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}(\Omega)$\end{document}Lp(Ω) for a domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega\subset\mathbb{R}^{N}$\end{document}Ω⊂RN (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\in\mathbb{N}$\end{document}N∈N), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq q\leq p\leq\infty$\end{document}1≤q≤p≤∞. We previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein’s extension operator. Although this formula can be applied to a domain Ω that can be divided into a finite number of Lipschitz domains, there was room for improvement in terms of accuracy. In this paper, we report that the accuracy of the embedding constant is significantly improved by restricting Ω to a domain dividable into bounded convex domains.


Introduction
We consider the Sobolev type embedding constant C p ( ) from W ,q ( ) ( ≤ q ≤ p ≤ ∞) to L p ( ). The constant C p ( ) satisfies for all u ∈ W ,q ( ), where ⊂ R N (N ∈ N) is a bounded domain and |x| = N j= x  j for x = (x  , . . . , x N ) ∈ R N . Here, L p ( ) ( ≤ p < ∞) is the functional space of the pth power Lebesgue integrable functions over endowed with the norm f L p ( ) := ( |f (x)| p dx) /p for f ∈ L p ( ), and L ∞ ( ) is the functional space of Lebesgue measurable functions over endowed with the norm f L ∞ ( ) = ess sup x∈ |f (x)| for f ∈ L ∞ ( ). Moreover, W k,p ( ) is the kth order L p -Sobolev space on endowed with the norm f W ,p ( ) = ( |f (x)| p dx + |∇f (x)| p dx) /p for f ∈ W ,p ( ) if  ≤ p < ∞ and f W ,∞ ( ) = ess sup x∈ |f (x)| + ess sup x∈ |∇f (x)| for f ∈ W ,∞ ( ) if p = ∞.
Since inequality () has significance for studies on partial differential equations, many researchers studied this type of Sobolev inequality and an explicit value of C p ( ) (see, e.g., [-]) following the pioneering work by Sobolev []. In particular, our interest is in the applicability of this constant to verified numerical computation methods for PDEs which originate from Nakao's [] and Plum's work []. These methods have been further developed by many researchers (see, e.g., [-] and the references therein).
The existence of C p ( ) for various domains (e.g., domains with the cone condition, domains with the Lipschitz boundary, and the (ε, δ)-domains) has been proven by constructing suitable extension operators from W k,p ( ) to W k,p (R N ) (see, e.g., [-]).
Several formulas for computing explicit values of C p ( ) have been proposed under suitable conditions. For example, the best constant in the classical Sobolev inequality on R N was independently shown by Aubin [] and Talenti []. For the case in which N =  and p = ∞, the best constant of C p ( ) was proposed under some boundary conditions, e.g., the Dirichlet, the Neumann, and the periodic condition [-]. For a square domain ⊂ R  , a tight estimate of C p ( ) was provided in []. Moreover, the best constant for the embed- where φ is the empty set and denotes the closure of (see Theorem .). Although this formula is applicable to such general domains, the values computed by this formula are very large; see Section  for concrete values.
In this paper, we report that the accuracy of the estimation of C p ( ) is significantly improved by restricting each i to bounded convex domain. Since any bounded convex domain is a Lipschitz domain (see, e.g., []), the present class of is somewhat special compared with the class treated in []. Nevertheless, the formulas presented in this paper still have applicability to various domains. To obtain a sharper estimation of C p ( ), we focus on the constants D p ( ) such that Here, | | is the measure of and u : → R is a constant function defined by x → u (x) = | | - u(y) dy. Inequality () is called the Sobolev-Poincaré inequality, and D p ( ) in () leads to the explicit value of C p ( ) (see Theorem .). Inequality () has also been studied by many researchers (see, e.g., [-]). For example, for a John domain , the existence of D p ( ) was shown while assuming that  ≤ q < N , p = Nq/(Nq) []. It was also shown that, when p = Nq/(Nq), D p ( ) exists if and only if W ,q ( ) is continuously embedded into L p ( ) []. Moreover, there are several formulas for obtaining an explicit value of D p ( ) for one-dimensional domains [-]. In the higher-dimensional cases, Table 1 The assumptions of p, q, and N imposed on Theorems 3.1, 3.2, 3.3, and 3.4 however, little is known about explicit values of D p ( ), except for some special cases (see, e.g., [] and [] for the cases in which p = q =  and p = q = , respectively). We propose four theorems (Theorem . to .) for obtaining explicit values of D p ( ) on a bounded convex domain . Each theorem can be used under the corresponding conditions listed in Table .
Theorems . and . are derived from the best constant in the Hardy-Littlewood-Sobolev inequality on R N . Theorems . and . are derived from the best constant in Young's inequality on R N . The values of D p ( ) calculated by these theorems yield the explicit values of C p ( ) combined with Theorem ..
The remainder of this paper is organized as follows. In Section , we propose Theorem . in which a formula for deriving an explicit value of C p ( ) from known D p ( ) is provided. In Section , we prove the four formulas (Theorems . to .) for obtaining the explicit values of D p ( ). In Section , we present examples where explicit values of C p ( ) are estimated for certain domains.

Estimation of embedding constant C p ( )
The following notation is used throughout this paper. For any bounded domain S ⊂ R N (N ∈ N), we define d S :=sup x,y∈S |x -y|. The closed ball centered around z ∈ R N with radius ρ >  is denoted by B(z, ρ) := {x ∈ R N | |x -z| ≤ ρ}. For m ≥ , let m be Hölder's conjugate of m, that is, m is defined by For two domains ⊆ R N and ⊆ R N such that ⊆ , we define the operator E , : In the following theorem, we provide a formula for obtaining an explicit value of C p ( ) from known D p ( ).
Theorem . Let ⊂ R N (N ∈ N) be a bounded domain, and let p and q satisfy  ≤ q ≤ p ≤ ∞. Suppose that there exists a finite number of bounded domains i (i = , , , . . . , n) satisfying () and (). Moreover, suppose that, for every i (i = , , , . . . , n), there exist Then () holds valid for where this formula is understood with /∞ =  when p = ∞ and/or q = ∞.
Proof Let u ∈ W ,q ( ). Since every i is bounded, Hölder's inequality states that We describe the following proof separately for the case of p = ∞ and p < ∞.

From () and (), it follows that
This implies that Theorem . holds for the case of p = ∞ and q = ∞. For q < ∞, we have where the last inequality follows from (s + t) q ≤  q- (s q + t q ) for s, t ≥ .
When p < ∞, we have From () and (), it follows that Therefore, we obtain

Estimation of D p ( i )
Let be the gamma function, that is, In the following three lemmas, we recall some known results required to obtain explicit values of D p ( i ) in () for bounded convex domains i .
A proof of Lemma . is provided in Appendix  because Lemma . plays an especially important role in obtaining the explicit values of D p ( i ).
holds valid for where this is the best constant in ().
holds valid for where this is the best constant in ().

Lemma . and () give
Theorem . Let ⊂ R N (N ∈ N) be a bounded convex domain, and let q > N . Then we have where V is defined in Theorem ..
Proof First, we show I := |x| -N q Therefore, where J is defined in the proof of Theorem .. Next, we prove ().
, where ψ is denoted in the proof of Theorem .. From Lemma . and (), for u ∈ W ,q ( ), it follows that

Explicit values of C p ( ) for certain domains
In

Estimation on a square domain
For the first example, we select the case in which = (, )  . For n = , , , , . . . , we define each i ( ≤ i ≤ n) as a square with side length / √ n; see Figure  for the cases in   which n =  and n = . For this division of , Theorem . states that In this case, V (in Theorems . and .) becomes a square with side length / √ n (see We also show the values of C ∞ ( ) computed by Theorem . for  ≤ q ≤  in Table .

Estimation on a triangle domain
For the second example, we select the case in which is a regular triangle with the vertices (, ), (, ), and (/, √ /). For n = , , , , . . . , we define each i ( ≤ i ≤ n) as a regular triangle with side length / √ n; see Figure  for the case in which n =  and n = . For this division of , Theorem . states that , max ≤i≤n D p ( i ) .
In this case, V is the regular hexagon displayed in Figure . Table      We also show the values of C ∞ ( ) computed by Theorem . for  ≤ q ≤  in Table .

Remark .
The values of C p ( ) derived from Theorem . to . (provided in Tables  to ) can be directly used for any domain that is composed of unit squares and triangles with side length  (see Figure  for some examples).

Estimation on a cube domain
For the third example, we select the case in which = (, )  . For n = , , , , . . . , we define each i ( ≤ i ≤ n) as a cube with side length /  √ n. For this division of , Theorem . states that In this case, V is also a cube with the side length /  √ n. Table  Table .

Conclusion
We proposed several theorems that provide explicit values of Sobolev type embedding constant C p ( ) satisfying () for a domain that can be divided into a finite number of bounded convex domains. These theorems give sharper estimates of C p ( ) than the previous estimates derived by the method in []. This accuracy improvement leads to much applicability of the estimates of C p ( ) to verified numerical computations for PDEs.

Appendix 1: Embedding constant C p ( ) on dividable domains
Theorem . provides an estimation of the embedding constant C p ( ) for a domain that can be divided into domains i (such as convex domains and Lipschitz domains) satisfying () and (). Proof We consider both the cases in which p < ∞ and p = ∞.
When p < ∞, it follows that Note that |x| p ≤ M p,q |x| q holds for x = (x  , x  , . . . , x n ) ∈ R n (see [, Lemma A.] for a detailed proof ), where we denote When p = ∞, Since M ∞,q = , we have u L ∞ ( ) ≤ max ≤i≤n C p ( i ) u W ,q ( ) .

Appendix 2: A proof of Lemma 3.1
This section provides a proof of Lemma . based on [, Lemma .].
Proof of Lemma . Since C ∞ ( ) ∩ W , ( ) is densely defined in W , ( ), it suffices to prove Lemma . for u ∈ C  ( ). Since is convex, we have, for x, y ∈ , where ω = (yx)/|y -x| and ∂ r u(x + rω) = ∂ ∂r u(x + rω). Integrating with respect to y over , we obtain Therefore, a proof of Lemma . is completed.